International Mathematical Forum, Vol. 9, 2014, no. 15, 725 - 731 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.4344
Convergence Theorems for Pettis Integral of Functions Taking Values in Locally Convex Spaces
Zamir Selko
Faculty of Natural Sciences Department of Mathematics University of Elbasan, Albania
Copyright c 2014 Zamir Selko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper we present convergence theorems for Pettis integral of functions defined on a complete probability space and taking values in a complete locally convex topological vector space.
Mathematics Subject Classification: 28B05, 46G10
Keywords: Pettis integral, convergence theorems, complete locally convex topological vector space
1 Introduction
We present convergence theorems for Pettis integral of functions defined on a complete probability space and taking values in a complete locally convex topological vector space, Theorems 2.2 and 2.3, which are the analogues of the known convergence theorems for Pettis integral of functions taking values in a Banach space. For that purpose, we apply a technique which is based on the limit projective of Banach spaces, Lemma 2.1. Theorem 2.2 is the analogue of Vitali Theorem for Pettis integral of func- tions taking values in a Banach space, Theorem 8.1 in [6]. Theorem 2.2 has been proved by another approach in [7]. Theorem 2.3 is the analogue of The- orem 2.8 in [9]. For more information on convergence theorems for Pettis integral, see [6], [7], [8], [4] and [2]. 726 Zamir Selko
Throughout this paper (Ω, Σ,μ) is a complete probability space and V is a complete locally convex space with its topology τ and topological dual V .By P we denote the family of all continuous semi-norms in this space; for every p ∈ P, V p denotes the quotient vector space of the vector space V with respect p to the equivalence relation x ∼p y ⇔ p(x − y) = 0; the map φp : V → V is the canonical quotient map, thus is the equivalence class of an element with p respect to the relation ”∼p”; the quotient normed space (V ,p) is called the normed component of the space V , where p(φp(x)) = p(x), for each x ∈ V ; p the Banach space (V , p) which is the completion of the space (V p,p) is called the Banach component of the space V ; Vp and V p are the topological duals of p (V p,p) and (V , p) respectively. It is easy to see that